Weighting Technique Implementation

Introduction

Problem

As it can be seen in 20_04_08_Ridge.html, there is a relation between the Probability the model assigns to a gene and the gene’s mean level of expression. This is a problem because we had previously discovered a bias in the SFARI scores related to mean level of expression (Preprocessing/Gandal/AllRegions/RMarkdowns/20_04_03_SFARI_genes.html), which means that this could be a confounding factor in our model and the reason why it seems to perform well, so we need to remove this bias to recover the true biological signal that is mixed with it and improve the quality of our model.

General idea

train model with equal weights for all samples

for l in loop:
  calculate bias
  correct weights to reduce bias
  retrain model
  
Return last model

Pseudocode

Parameters:

eta: Learning rate
T: Number of loops
D: Training data
H: Classification model
c: bias constraint
lambda: scaling factor for the weights
$w_i$ with $i=1,...,N$: Weights assigned to each sample

Pseudocode:

lambda = 0
w = [1, ..., 1]
c = std(meanExpr(D))

h  = train classifier H with lambda and w

for t in 1,,,T do
  bias = <h(x), c(x)>
  update lambda to lambda - eta*bias
  update weights_hat to exp(lambda*mean(c))
  update weights to w_hat/(1+w_hat) if y_i=1, 1/(1+w_hat) if y_i=0
  update h with new weights
  
Return h

Remove Bias

Demographic Parity as a measure of Bias

Using Demographic Parity as a measure of bias: A fair classifier h should make positive predictions each segment $G$ of the population at the same rate as in all of the population

This definition is for discrete segments of the population. Since our bias is found across all the population but in different measures depending on the mean level of expression of the gene, we have to adapt this definition to a continuous bias scenario

Demographic Parity for our problem: A fair classifier h should make positive predictions on genes with a certail mean level of expression at the same rate as in all of the genes in the dataset

Bias Metric

The original formula for the Demographic Parity bias is

$c(x,0) = 0 $ when the prediction is negative
$c(x,1) = \frac{g(x)}{Z_G}-1$ when the prediction is positive. Where $g(x)$ is the Kronecker delta to indicate if the sample belongs to the protected group and $Z_G$ is the proportion of the population that belongs to the group we want to protect against bias

Using this definitions in our problem:

$g(x):$ Since all our samples belong to the protected group, this would always be 1

$Z_G:$ Since all of our samples belong to the protected group, this would also always be 1

So our measure of bias $c(x,1) = \frac{1}{1}-1 = 0$ for all samples. This doesn’t work, so we need to adapt it to our continous case

Adaptation of the bias metric

We can use $c(x,1) = std(meanExpr(x))$ as the constraint function, this way, when we calculate the bias of the dataset:

$h(x)\cdot c(x)$ will only be zero if the positive samples are balanced around the mean expression, and the sign of the bias will indicate the direction of the bias

Calculating the Weights

Notes:

This model is only going to be used to obtain the optimal weights for the observations. The weights obtained in this model are going to be the ones used to train the final model afterwards
In the original model we would use oversampling as part of the training function, but here, since we are adding weights to the samples, the oversampling will have to be performed separately (otherwise the weights don’t match the number of observations), so it was moved inside the create_train_test_sets function

### DEFINE FUNCTIONS

create_train_test_sets = function(p, seed){
  
  # Get SFARI Score of all the samples so our train and test sets are balanced for each score
  sample_scores = dataset %>% mutate(ID = rownames(.)) %>% dplyr::select(ID) %>% 
                  left_join(biased_predictions %>% dplyr::select(ID, gene.score), by = 'ID') %>% 
                  mutate(gene.score = ifelse(is.na(gene.score), 'None', gene.score))

  set.seed(seed)
  train_idx = createDataPartition(sample_scores$gene.score, p = p, list = FALSE)
  train_set = dataset[train_idx,]
  test_set = dataset[-train_idx,]
  
  # Modify SFARI label in train set, save gene IDS (bc we lose them with SMOTE) and perform oversampling using SMOTE
  train_set = train_set %>% mutate(SFARI = ifelse(SFARI == TRUE, 'SFARI', 'not_SFARI') %>% as.factor,
                                   ID = rownames(.) %>% as.factor) %>% SMOTE(form = SFARI ~ . - ID)
  train_set_IDs = train_set %>% pull(ID)
  
  return(list('train_set' = train_set %>% dplyr::select(-ID), 'test_set' = test_set, 
              'train_set_IDs' = train_set_IDs))
}

run_weights_model = function(p, seed, Loops){
  
  # CREATE TRAIN AND TEST SETS
  train_test_sets = create_train_test_sets(p, seed)
  train_set = train_test_sets[['train_set']]
  test_set = train_test_sets[['test_set']]
  train_set_IDs = train_test_sets[['train_set_IDs']]
  
  
  # SET INITIAL PARAMETERS
  
  # General parameters
  set.seed(seed)
  lambda_seq = 10^seq(1, -4, by = -.1)
  k_fold = 10
  cv_repeats = 5
  trControl = trainControl(method = 'repeatedcv', number = k_fold, repeats = cv_repeats, verboseIter = FALSE, 
                           classProbs = TRUE, savePredictions = 'final', summaryFunction = twoClassSummary)
  # Bias correction parameters
  eta = 0.5
  lambda = 0
  w = rep(1, nrow(train_set))
  
  
  # TRAIN MODEL
  h = train(SFARI ~., data = train_set, method = 'glmnet', trControl = trControl, metric = 'ROC',
            tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  
  
  # CORRECT BIAS
  
  # Mean Expression info
  mean_expr = data.frame('ID' = train_set_IDs) %>% 
              left_join(data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)), by = 'ID') %>%
              mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))
  
  # Track behaviour of plot
  bias_vec = c()
  acc_vec = c()
  
  for(l in 1:Loops){
    
    # Calculate bias for positive predicted samples
    bias = mean(mean_expr$meanExpr_std[predict(h, train_set)=='SFARI'])
    
    # Update weights
    lambda = lambda - eta*bias
    w_hat = exp(lambda*mean_expr$meanExpr_std)
    w = 1/(1+w_hat)
    w[train_set$SFARI=='SFARI'] = w[train_set$SFARI=='SFARI']*w_hat[train_set$SFARI=='SFARI']
    
    # Update tracking vars
    bias_vec = c(bias_vec, bias)
    acc_vec = c(acc_vec, mean(predict(h, train_set) == train_set$SFARI))
    
    # Update h
    h = train(SFARI ~., data = train_set, method = 'glmnet', weights = w, trControl = trControl, 
              metric = 'ROC', tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  }

  
  return(list('lambda' = lambda, 'bias_vec' = bias_vec, 'acc_vec' = acc_vec))
}


### RUN MODEL TO FIND OPTIMAL WEIGHTS

# Parameters
p = 0.75
seed = 123
Loops = 50

# Run model
model_output = run_weights_model(p, seed, Loops)

# Extract metrics
lambda = model_output[['lambda']]
bias_vec = model_output[['bias_vec']]
acc_vec = model_output[['acc_vec']]


rm(p, seeds, Loops, run_weights_model)

The optimal value of $\lambda$ is -1.4970506

The bias decreases until it oscilates around zero and the accuracy is not affected much

plot_info = data.frame('iter' = 1:length(bias_vec), 'bias' = bias_vec, 'accuracy' = acc_vec) %>% 
            melt(id.vars = 'iter')

plot_info %>% ggplot(aes(x=iter, y=value, color = variable)) + geom_line() + xlab('Iteration') + theme_minimal()

Since the bias increases the probability of being classified as 1 for genes with higher levels of expression, as the level of expression of a gene increases, the algorithm:

Increases the weight of genes with a negative label
Decreases the weight of genes with a positive label

mean_expr = data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)) %>%
            left_join(predictions, by = 'ID') %>% filter(n>0) %>%
            mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))

w_hat = exp(lambda*mean_expr$meanExpr_std) # inverso a mean expr
w0 = 1/(1+w_hat) # prop a mean expr
w = 1/(1+w_hat)
w[mean_expr$SFARI %>% as.logical] = w[mean_expr$SFARI %>% as.logical]*w_hat[mean_expr$SFARI %>% as.logical] # inv mean expr Positives, prop Negatives
plot_data = data.frame('meanExpr' = mean_expr$meanExpr, 'w_hat' = w_hat, 'w0' = w0, 'w' = w, 
                       'SFARI' = mean_expr$SFARI, 'pred' = mean_expr$pred)

plot_data %>% ggplot(aes(meanExpr, w, color = SFARI)) + geom_point(alpha = 0.3) + 
              xlab('Mean Expression') + ylab('Weight') +
              ggtitle('Weights of the final model') + ylim(c(0,1)) + theme_minimal()

rm(mean_expr, w_hat, w0, w)

Running the final model

run_final_model = function(p, seed, lambda){
  
  # CREATE TRAIN AND TEST SETS
  train_test_sets = create_train_test_sets(p, seed)
  train_set = train_test_sets[['train_set']]
  test_set = train_test_sets[['test_set']]
  train_set_IDs = train_test_sets[['train_set_IDs']]
  
  
  # SET INITIAL PARAMETERS
  
  # General parameters
  set.seed(seed)
  lambda_seq = 10^seq(1, -4, by = -.1)
  k_fold = 10
  cv_repeats = 5
  trControl = trainControl(method = 'repeatedcv', number = k_fold, repeats = cv_repeats, verboseIter = FALSE, 
                           classProbs = TRUE, savePredictions = 'final', summaryFunction = twoClassSummary)
  
  # Bias correcting parameters
  mean_expr = data.frame('ID' = train_set_IDs) %>% 
              left_join(data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)), by = 'ID') %>%
              mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))
  w_hat = exp(lambda*mean_expr$meanExpr_std)
  w = 1/(1+w_hat)
  w[train_set$SFARI=='SFARI'] = w[train_set$SFARI=='SFARI']*w_hat[train_set$SFARI=='SFARI']
  
  
  # TRAIN MODEL
  fit = train(SFARI ~., data = train_set, method = 'glmnet', weights = w, trControl = trControl, 
              metric = 'ROC', tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  
  
  # PREDICT TEST SET LABELS AND CREATE PERFORMANCE METRICS
  
  # Predict labels in test set
  predictions = fit %>% predict(test_set, type = 'prob')
  preds = data.frame('ID' = rownames(test_set), 'corrected_prob' = predictions$SFARI) %>% 
          mutate(corrected_pred = corrected_prob > 0.5)
  

  # Measure performance of the model
  acc = mean(test_set$SFARI==preds$corrected_pred)
  prec = Precision(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  rec = Recall(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  F1 = F1_Score(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  pred_ROCR = prediction(preds$corrected_prob, test_set$SFARI)
  AUC = performance(pred_ROCR, measure='auc')@y.values[[1]]
  
  # Extract coefficients from features
  coefs = coef(fit$finalModel, fit$bestTune$lambda) %>% as.vector
  
  
  return(list('acc' = acc, 'prec' = prec, 'rec' = rec, 'F1' = F1, 'AUC' = AUC, 'preds' = preds, 'coefs'= coefs))
}


### RUN MODEL

# Parameters
p = 0.75
n_iter = 25
seeds = 123:(123+n_iter-1)

# So the input is the same as in 10_classification_model.html
original_dataset = dataset %>% mutate(ID = rownames(.)) %>% 
                   left_join(biased_predictions %>% dplyr::select(ID, gene.score))

# Store outputs
acc = c()
prec = c()
rec = c()
F1 = c()
AUC = c()
predictions = data.frame('ID' = rownames(dataset), 'SFARI' = dataset$SFARI, 'corrected_prob' = 0, 
                         'corrected_pred' = 0, 'n' = 0)
coefs = data.frame('var' = c('Intercept', colnames(dataset[,-ncol(dataset)])), 'coef' = 0)

for(seed in seeds){
  
  # Run model
  model_output = run_final_model(p, seed, lambda)
  
  # Update outputs
  acc = c(acc, model_output[['acc']])
  prec = c(prec, model_output[['prec']])
  rec = c(rec, model_output[['rec']])
  F1 = c(F1, model_output[['F1']])
  AUC = c(AUC, model_output[['AUC']])
  preds = model_output[['preds']]
  coefs$coef = coefs$coef + model_output[['coefs']]
  update_preds = preds %>% dplyr::select(-ID) %>% mutate(n=1)
  predictions[predictions$ID %in% preds$ID, c('corrected_prob','corrected_pred','n')] = 
    predictions[predictions$ID %in% preds$ID, c('corrected_prob','corrected_pred','n')] +
     update_preds
}

coefs = coefs %>% mutate(coef = coef/n_iter)
predictions = predictions %>% mutate(corrected_prob = corrected_prob/n, corrected_pred_count = corrected_pred, 
                                     corrected_pred = corrected_prob>0.5) %>% 
              left_join(biased_predictions %>% dplyr::select(ID, prob, pred), by = 'ID')


rm(p, seeds, update_preds, create_train_test_sets, run_final_model)

test_set = predictions %>% filter(n>0) %>% 
           left_join(dataset %>% mutate(ID = rownames(.)) %>% dplyr::select(ID, GS, MTcor), by = 'ID')
rownames(test_set) = predictions$ID[predictions$n>0]

Results

The relation between the model probability and the mean level of expression of the genes is not completely gone, there seems to be a negative relation for the genes with the lowest levels of expression

Even though the trend line is not as flat as with the first method, we are not fixing this directly as we were doing before, this is now just a consequence of the corrections we did inside of the model, so it makes sense for it to be less exact than before

# Plot results
plot_data = data.frame('ID'=rownames(datExpr), 'meanExpr'=rowMeans(datExpr)) %>% 
            right_join(test_set, by='ID')

plot_data %>% ggplot(aes(meanExpr, corrected_prob)) + geom_point(alpha=0.2, color='#0099cc') +
              geom_smooth(method='gam', color='gray', alpha=0.2) + 
              xlab('Mean Expression') + ylab('Corrected Probability') +
              theme_minimal() + ggtitle('Mean expression vs Model Probability corrected using adjusted Weights')

Performance metrics

Confusion matrix

conf_mat = test_set %>% apply_labels(SFARI = 'Actual Labels', 
                                     corrected_prob = 'Assigned Probability', 
                                     corrected_pred = 'Label Prediction')

cro(conf_mat$SFARI, list(conf_mat$corrected_pred, total()))

	Label Prediction		#Total
	FALSE	TRUE
Actual Labels
FALSE	11588	967	12555
TRUE	562	78	640
#Total cases	12150	1045	13195

rm(conf_mat)

Accuracy: Mean = 0.8554 SD = 0.0354

Precision: Mean = 0.0722 SD = 0.0184

Recall: Mean = 0.1617 SD = 0.0515

F1 score: Mean = 0.0969 SD = 0.0228

ROC Curve: Mean = 0.5636 SD = 0.0255

pred_ROCR = prediction(test_set$corrected_prob, test_set$SFARI)

roc_ROCR = performance(pred_ROCR, measure='tpr', x.measure='fpr')
auc = performance(pred_ROCR, measure='auc')@y.values[[1]]

plot(roc_ROCR, main=paste0('ROC curve (AUC=',round(mean(AUC),2),')'), col='#009999')
abline(a=0, b=1, col='#666666')

Lift Curve

lift_ROCR = performance(pred_ROCR, measure='lift', x.measure='rpp')
plot(lift_ROCR, main='Lift curve', col='#86b300')

rm(pred_ROCR, roc_ROCR, AUC, lift_ROCR, acc, acc_vec, auc, bias_vec, F1, prec, rec)

Coefficients

gene_corr_info = dataset %>% mutate('ID' = rownames(dataset)) %>% dplyr::select(ID, MTcor, SFARI) %>% 
                 left_join(assigned_module, by ='ID') %>% mutate(Module = gsub('#','',Module))

coef_info = coefs %>% mutate('feature' = gsub('MM.','',var)) %>% 
            left_join(gene_corr_info, by = c('feature' = 'Module')) %>% 
            dplyr::select(feature, coef, MTcor, SFARI) %>% group_by(feature, coef, MTcor) %>% 
            summarise('SFARI_perc' = mean(SFARI)) %>% arrange(desc(coef))

coef_info %>% dplyr::select(feature, coef) %>% filter(feature %in% c('Intercept','GS','absGS','MTcor')) %>%
              dplyr::rename('Feature' = feature, 'Coefficient' = coef) %>% 
              kable(align = 'cc', caption = 'Regression Coefficients') %>% kable_styling(full_width = F)

Regression Coefficients
Feature	Coefficient
MTcor	0.0231677
GS	-0.0201093
absGS	-0.0209015
Intercept	-0.4946006

There is still a positive relation between the coefficient assigned to the membership of each module and the enrichment (using ORA) in SFARI genes that are assigned to that module

load('./../Data/ORA.RData')

enrichment_SFARI_info = data.frame('Module'=as.character(), 'SFARI_enrichment'=as.numeric())
for(m in names(enrichment_SFARI)){
  m_info = enrichment_SFARI[[m]]
  enrichment = 1-ifelse('SFARI' %in% m_info$ID, m_info$pvalue[m_info$ID=='SFARI'],1)
  enrichment_SFARI_info = enrichment_SFARI_info %>% 
                          add_row(Module = gsub('#','',m), SFARI_enrichment = enrichment)
}

plot_data = coef_info %>% dplyr::rename('Module' = feature) %>% 
            left_join(enrichment_SFARI_info, by = 'Module') %>% filter(!is.na(MTcor))

ggplotly(plot_data %>% ggplot(aes(coef, SFARI_enrichment)) + 
         geom_smooth(method = 'lm', color = 'gray', alpha = 0.1) + 
         geom_point(aes(id = Module), color = paste0('#',plot_data$Module), alpha=0.7) + 
         theme_minimal() + xlab('Coefficient') + 
         ylab('SFARI Genes Enrichment'))

rm(enrichment_old_SFARI, enrichment_DGN, enrichment_DO, enrichment_GO, enrichment_KEGG, enrichment_Reactome, m,
   m_info, enrichment)

ggplotly(coef_info %>% dplyr::rename('Module' = feature) %>% filter(!is.na(MTcor)) %>%
         ggplot(aes(coef, MTcor)) +  geom_smooth(method = 'lm', color = 'gray', alpha = 0.1) + 
         geom_point(aes(id = Module), color=paste0('#',coef_info$feature[!is.na(coef_info$MTcor)]), alpha=.7) + 
         theme_minimal() + xlab('Coefficient') + ylab('Module-Diagnosis correlation'))

Analyse Results

Probability distribution by SFARI Label

SFARI genes have a higher Probability distribution than the rest, but the overlap is larger than before

plot_data = test_set %>% dplyr::select(corrected_prob, SFARI)

ggplotly(plot_data %>% ggplot(aes(corrected_prob, fill=SFARI, color=SFARI)) + geom_density(alpha=0.3) + 
         geom_vline(xintercept = mean(plot_data$corrected_prob[plot_data$SFARI]), color = '#00C0C2', 
                    linetype='dashed') +
         geom_vline(xintercept = mean(plot_data$corrected_prob[!plot_data$SFARI]), color = '#FF7371', 
                    linetype='dashed') +
        xlab('Score') + ggtitle('Model Probability distribution by SFARI Label') + theme_minimal())

Probability distribution by SFARI Gene Scores

The relation between probability and SFARI Gene Scores weakened but it’s still there

plot_data = test_set %>% mutate(ID=rownames(test_set)) %>% dplyr::select(ID, corrected_prob) %>%
            left_join(original_dataset, by='ID') %>% dplyr::select(ID, corrected_prob, gene.score) %>% 
            apply_labels(gene.score='SFARI Gene score')

cro(plot_data$gene.score)

	#Total
SFARI Gene score
1	105
2	168
3	364
Neuronal	782
Others	11762
#Total cases	13181

mean_vals = plot_data %>% group_by(gene.score) %>% summarise(mean_prob = mean(corrected_prob))

comparisons = list(c('1','2'), c('2','3'), c('3','Neuronal'), c('Neuronal','Others'),
                   c('1','3'), c('3','Others'), c('2','Neuronal'),
                   c('1','Neuronal'), c('2','Others'), c('1','Others'))
increase = 0.07
base = 0.75
pos_y_comparisons = c(rep(base, 4), rep(base + increase, 2), base + 2:5*increase)

plot_data %>% filter(!is.na(gene.score)) %>% ggplot(aes(gene.score, corrected_prob, fill=gene.score)) + 
              geom_boxplot(outlier.colour='#cccccc', outlier.shape='o', outlier.size=3) +
              stat_compare_means(comparisons = comparisons, label = 'p.signif', method = 't.test', 
                                 method.args = list(var.equal = FALSE), label.y = pos_y_comparisons, 
                                 tip.length = .02) +
              scale_fill_manual(values=SFARI_colour_hue(r=c(1:3,8,7))) + 
              ggtitle('Distribution of probabilities by SFARI score') +
              xlab('SFARI score') + ylab('Probability') + theme_minimal() + theme(legend.position = 'none')

rm(mean_vals, increase, base, pos_y_comparisons)

Genes with the highest Probabilities

The concentration of SFARI genes decrease from 1:4 to 1:12
The genes with the highest probabilities are no longer SFARI Genes

test_set %>% dplyr::select(corrected_prob, SFARI) %>% mutate(ID = rownames(test_set)) %>% 
             arrange(desc(corrected_prob)) %>% top_n(50, wt=corrected_prob) %>%
             left_join(biased_predictions %>% dplyr::select(ID, gene.score, external_gene_id, MTcor, GS), 
                       by = 'ID') %>%
             dplyr::rename('GeneSymbol' = external_gene_id, 'Probability' = corrected_prob, 
                           'ModuleDiagnosis_corr' = MTcor, 'GeneSignificance' = GS) %>%
             mutate(ModuleDiagnosis_corr = round(ModuleDiagnosis_corr,4), Probability = round(Probability,4), 
                    GeneSignificance = round(GeneSignificance,4)) %>%
             left_join(assigned_module, by = 'ID') %>%
             dplyr::select(GeneSymbol, GeneSignificance, ModuleDiagnosis_corr, Module, Probability,
                           gene.score) %>%
             kable(caption = 'Genes with highest model probabilities from the test set') %>% 
             kable_styling(full_width = F)

Genes with highest model probabilities from the test set
GeneSymbol	GeneSignificance	ModuleDiagnosis_corr	Module	Probability	gene.score
TRPC6	0.1888	0.4910	#E28900	0.6947	2
PLXDC2	0.3497	0.5272	#F564E4	0.6781	Others
DIP2C	-0.1778	0.2573	#FA7377	0.6654	2
MB21D2	-0.0983	-0.0793	#63B200	0.6601	Others
CPT1C	-0.0969	-0.0102	#00BDD3	0.6448	Others
PCDH15	0.1827	0.3849	#C89800	0.6397	3
TMEM184A	0.1764	0.0319	#00B6EA	0.6371	Others
CMTM7	-0.1504	-0.0683	#FF6B93	0.6338	Others
FBXL13	0.1544	0.5621	#4BB400	0.6325	Others
GABRA5	0.5898	0.4329	#00C0BF	0.6319	Neuronal
OCA2	-0.2946	-0.3932	#FE6E8A	0.6314	Others
RPTOR	0.3295	0.1819	#DC8D00	0.6299	Others
PRKCE	-0.0371	0.4910	#E28900	0.6288	Others
CSF2RA	-0.3589	-0.4038	#00B4EF	0.6285	Others
RPRD2	0.1194	0.3738	#FC7181	0.6264	Others
FA2H	-0.2586	-0.2919	#00BFC3	0.6263	Others
OR51I1	0.1205	0.4910	#E28900	0.6256	Others
TMEM211	-0.4070	-0.4513	#FF699C	0.6249	Others
CACNA1D	-0.4023	-0.2919	#00BFC3	0.6226	2
PLXNC1	0.5660	0.5125	#89AC00	0.6217	Others
TUSC3	0.1125	-0.0102	#00BDD3	0.6217	Others
MYRF	-0.3458	-0.2919	#00BFC3	0.6214	Others
GABBR2	0.1204	0.2418	#A68AFF	0.6187	3
IQCF3	0.4313	0.6982	#FF68A0	0.6167	Others
CACNA1C	-0.1527	0.2418	#A68AFF	0.6164	1
IL4R	-0.1025	-0.1266	#00BA3B	0.6139	Others
IFI16	-0.4044	-0.6465	#00C0B8	0.6138	Others
ANKRD2	0.0212	-0.0683	#FF6B93	0.6109	Others
ACAP1	-0.3687	-0.1564	#C39A00	0.6082	Others
OPRM1	0.3906	0.4910	#E28900	0.6081	Others
MX2	-0.3168	-0.0728	#DE8C00	0.6074	Others
KIAA1644	-0.0078	-0.1197	#9EA700	0.6071	Others
WIPF1	-0.1209	0.0860	#96A900	0.6069	Others
FAM98B	-0.2066	-0.3187	#D39300	0.6069	Others
CD86	-0.4779	-0.6465	#00C0B8	0.6068	Others
CHST8	-0.0678	-0.5295	#C69900	0.6056	Others
C14orf39	-0.1211	0.0270	#00B7E7	0.6052	Others
GRIA1	0.2054	0.4910	#E28900	0.6047	2
IL1B	-0.6507	-0.8841	#E76CF3	0.6042	Others
LRRC16B	0.0365	-0.0793	#63B200	0.6032	Others
CREBBP	-0.0191	0.0401	#00BF7F	0.6025	1
POLR1E	-0.3134	-0.4513	#FF699C	0.6024	Others
C1orf95	-0.2008	0.1945	#D59100	0.6024	Others
SHISA9	-0.0285	0.2203	#F17D50	0.6020	Neuronal
RAE1	0.3609	0.4329	#00C0BF	0.6013	Others
CECR1	0.2609	0.2029	#7AAE00	0.6013	Others
CHST1	0.2286	0.4910	#E28900	0.6005	Others
PLXNA1	0.2434	0.2418	#A68AFF	0.6005	Others
TP53I11	-0.1272	0.5125	#89AC00	0.6001	Others
PRELP	0.2110	0.5125	#89AC00	0.6000	Others

Negative samples distribution

The objective of this model is to identify candidate SFARI genes. For this, we are going to focus on the negative samples (the non-SFARI genes)

negative_set = test_set %>% filter(!SFARI)

negative_set_table = negative_set %>% apply_labels(corrected_prob = 'Assigned Probability', 
                                                   corrected_pred = 'Label Prediction')

cro(negative_set_table$corrected_pred)

	#Total
Label Prediction
FALSE	11588
TRUE	967
#Total cases	12555

967 genes are predicted as ASD-related

negative_set %>% ggplot(aes(corrected_prob)) + geom_density(color='#F8766D', fill='#F8766D', alpha=0.5) +
                 geom_vline(xintercept=0.5, color='#333333', linetype='dotted') + xlab('Probability') +
                 ggtitle('Probability distribution of the Negative samples in the Test Set') + 
                 theme_minimal()

negative_set %>% dplyr::select(corrected_prob, SFARI) %>% mutate(ID = rownames(negative_set)) %>% 
                 arrange(desc(corrected_prob)) %>% top_n(50, wt=corrected_prob) %>%
                 left_join(biased_predictions %>% dplyr::select(ID, gene.score, external_gene_id, MTcor, GS), 
                           by = 'ID') %>%
                 dplyr::rename('GeneSymbol' = external_gene_id, 'Probability' = corrected_prob, 
                               'ModuleDiagnosis_corr' = MTcor, 'GeneSignificance' = GS) %>%
                 mutate(ModuleDiagnosis_corr = round(ModuleDiagnosis_corr,4), 
                        Probability = round(Probability,4), 
                        GeneSignificance = round(GeneSignificance,4)) %>%
                 left_join(assigned_module, by = 'ID') %>%
                 dplyr::select(GeneSymbol, GeneSignificance, ModuleDiagnosis_corr, Module, Probability,
                               gene.score) %>%
                 kable(caption = 'Genes with highest model probabilities from the Negative set') %>% 
                 kable_styling(full_width = F)

Genes with highest model probabilities from the Negative set
GeneSymbol	GeneSignificance	ModuleDiagnosis_corr	Module	Probability	gene.score
PLXDC2	0.3497	0.5272	#F564E4	0.6781	Others
MB21D2	-0.0983	-0.0793	#63B200	0.6601	Others
CPT1C	-0.0969	-0.0102	#00BDD3	0.6448	Others
TMEM184A	0.1764	0.0319	#00B6EA	0.6371	Others
CMTM7	-0.1504	-0.0683	#FF6B93	0.6338	Others
FBXL13	0.1544	0.5621	#4BB400	0.6325	Others
GABRA5	0.5898	0.4329	#00C0BF	0.6319	Neuronal
OCA2	-0.2946	-0.3932	#FE6E8A	0.6314	Others
RPTOR	0.3295	0.1819	#DC8D00	0.6299	Others
PRKCE	-0.0371	0.4910	#E28900	0.6288	Others
CSF2RA	-0.3589	-0.4038	#00B4EF	0.6285	Others
RPRD2	0.1194	0.3738	#FC7181	0.6264	Others
FA2H	-0.2586	-0.2919	#00BFC3	0.6263	Others
OR51I1	0.1205	0.4910	#E28900	0.6256	Others
TMEM211	-0.4070	-0.4513	#FF699C	0.6249	Others
PLXNC1	0.5660	0.5125	#89AC00	0.6217	Others
TUSC3	0.1125	-0.0102	#00BDD3	0.6217	Others
MYRF	-0.3458	-0.2919	#00BFC3	0.6214	Others
IQCF3	0.4313	0.6982	#FF68A0	0.6167	Others
IL4R	-0.1025	-0.1266	#00BA3B	0.6139	Others
IFI16	-0.4044	-0.6465	#00C0B8	0.6138	Others
ANKRD2	0.0212	-0.0683	#FF6B93	0.6109	Others
ACAP1	-0.3687	-0.1564	#C39A00	0.6082	Others
OPRM1	0.3906	0.4910	#E28900	0.6081	Others
MX2	-0.3168	-0.0728	#DE8C00	0.6074	Others
KIAA1644	-0.0078	-0.1197	#9EA700	0.6071	Others
WIPF1	-0.1209	0.0860	#96A900	0.6069	Others
FAM98B	-0.2066	-0.3187	#D39300	0.6069	Others
CD86	-0.4779	-0.6465	#00C0B8	0.6068	Others
CHST8	-0.0678	-0.5295	#C69900	0.6056	Others
C14orf39	-0.1211	0.0270	#00B7E7	0.6052	Others
IL1B	-0.6507	-0.8841	#E76CF3	0.6042	Others
LRRC16B	0.0365	-0.0793	#63B200	0.6032	Others
POLR1E	-0.3134	-0.4513	#FF699C	0.6024	Others
C1orf95	-0.2008	0.1945	#D59100	0.6024	Others
SHISA9	-0.0285	0.2203	#F17D50	0.6020	Neuronal
RAE1	0.3609	0.4329	#00C0BF	0.6013	Others
CECR1	0.2609	0.2029	#7AAE00	0.6013	Others
CHST1	0.2286	0.4910	#E28900	0.6005	Others
PLXNA1	0.2434	0.2418	#A68AFF	0.6005	Others
TP53I11	-0.1272	0.5125	#89AC00	0.6001	Others
PRELP	0.2110	0.5125	#89AC00	0.6000	Others
HHLA1	0.2522	-0.0470	#00C1A1	0.5999	Others
LHFPL2	-0.1698	-0.0102	#00BDD3	0.5995	Others
PAN3	-0.2159	-0.5696	#FD6F86	0.5991	Others
SEC16A	0.0131	0.3738	#FC7181	0.5989	Others
DLG5	0.2711	0.9122	#ADA200	0.5987	Others
PXN	-0.0361	0.0961	#FD61D1	0.5982	Others
SUV39H1	-0.2674	-0.4513	#FF699C	0.5978	Others
YLPM1	0.1645	-0.0235	#FF62BC	0.5973	Others

Comparison with the original model’s probabilities:

The genes with the highest Probabilitiess were affected the most as a group
In general genes with the lowest Probabilities got their score increased and the genes with the highest scores, decreased
The change in Probability by gene is much larger than with the Post Processing approach

negative_set %>% mutate(diff = abs(prob-corrected_prob)) %>% 
             ggplot(aes(prob, corrected_prob, color = diff)) + geom_point(alpha=0.3) + scale_color_viridis() + 
             geom_abline(slope=1, intercept=0, color='gray', linetype='dashed') + 
             geom_smooth(color='#666666', alpha=0.5, se=TRUE, size=0.5) + coord_fixed() +
             xlab('Original probability') + ylab('Corrected probability') + theme_minimal() + theme(legend.position = 'none')

negative_set_table = negative_set %>% apply_labels(corrected_prob = 'Corrected Probability', 
                                                   corrected_pred = 'Corrected Class Prediction',
                                                   pred = 'Original Class Prediction') %>%
                     filter(!is.na(pred))

cro(negative_set_table$pred, list(negative_set_table$corrected_pred, total()))

	Corrected Class Prediction		#Total
	FALSE	TRUE
Original Class Prediction
FALSE	9809	286	10095
TRUE	1770	679	2449
#Total cases	11579	965	12544

84% of the genes maintained their original predicted class

rm(negative_set_table)

Probability and Gene Significance

The relation is the opposite as before, the higher the Gene Significance, the lower the probability, with the highest probabilities corresponding to under-expressed genes

*The transparent verison of the trend line is the original trend line

negative_set %>% ggplot(aes(corrected_prob, GS, color=MTcor)) + geom_point() + 
                 geom_smooth(method='gam', color='#666666') + ylab('Gene Significance') +
                 geom_line(stat='smooth', method='gam', color='#666666', alpha=0.5, size=1.2, aes(x=prob)) +
                 geom_hline(yintercept=mean(negative_set$GS), color='gray', linetype='dashed') +
                 scale_color_gradientn(colours=c('#F8766D','white','#00BFC4')) + xlab('Corrected Probability') +
                 ggtitle('Relation between the Model\'s Corrected Probability and Gene Significance') + 
                 theme_minimal()

Summarised version of Probability vs mean expression, plotting by module instead of by gene

The difference in the trend lines between this plot and the one above is that the one above takes all the points into consideration while this considers each module as an observation by itself, so the top one is strongly affected by big modules and the bottom one treats all modules the same

The transparent version of each point and trend lines are the original values and trends before the bias correction

plot_data = negative_set %>% mutate(ID = rownames(.)) %>% left_join(assigned_module, by = 'ID') %>%
            group_by(MTcor, Module) %>% summarise(mean = mean(prob), sd = sd(prob),
                                                  new_mean = mean(corrected_prob),
                                                  new_sd = sd(corrected_prob), n = n()) %>%
            mutate(MTcor_sign = ifelse(MTcor>0, 'Positive', 'Negative')) %>% 
            dplyr::select(Module, MTcor, MTcor_sign, mean, new_mean, sd, new_sd, n) %>% distinct()
colnames(plot_data)[1] = 'ID'

ggplotly(plot_data %>% ggplot(aes(MTcor, new_mean, size=n, color=MTcor_sign)) + geom_point(aes(id = ID)) + 
         geom_smooth(method='loess', color='gray', se=FALSE) + geom_smooth(method='lm', se=FALSE) + 
         geom_point(aes(y=mean), alpha=0.3) + 
         geom_line(stat='smooth', method='loess', color='gray', se=FALSE, alpha=0.3, size=1.2, aes(y=mean)) + 
         geom_line(stat='smooth', method='lm', se=FALSE, alpha=0.3, size=1.2, aes(y=mean)) + 
         xlab('Module-Diagnosis correlation') + ylab('Mean Corrected Probability by Module') + 
         theme_minimal() + theme(legend.position='none'))

Probability and mean level of expression

To check if correcting by gene also corrected by module: Yes, the bias seems to be removed completely

mean_and_sd = data.frame(ID=rownames(datExpr), meanExpr=rowMeans(datExpr), sdExpr=apply(datExpr,1,sd))

plot_data = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>% 
            left_join(mean_and_sd, by='ID') %>% 
            left_join(assigned_module, by='ID')

plot_data2 = plot_data %>% group_by(Module) %>% summarise(meanExpr = mean(meanExpr), meanProb = mean(prob), 
                                                          new_meanProb = mean(corrected_prob), n=n())

ggplotly(plot_data2 %>% ggplot(aes(meanExpr, new_meanProb, size=n)) + 
         geom_point(color=plot_data2$Module) + geom_point(color=plot_data2$Module, alpha=0.3, aes(y=meanProb)) + 
         geom_smooth(method='loess', se=TRUE, color='gray', alpha=0.1, size=0.7) + 
         geom_line(stat='smooth', method='loess', se=TRUE, color='gray', alpha=0.4, size=1.2, aes(y=meanProb)) +
         xlab('Mean Expression') + ylab('Corrected Probability') +  
         ggtitle('Mean expression vs corrected Model Probability by Module') +
         theme_minimal() + theme(legend.position='none'))

rm(plot_data2, mean_and_sd)

Probability and LFC

Unider-Expressed genes got their probabilities increased and over-expressed genes decreased

plot_data = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>% 
            left_join(DE_info %>% data.frame %>% mutate(ID=rownames(.)), by='ID') %>%
            dplyr::rename('log2FoldChange' = logFC, 'padj' = adj.P.Val)

plot_data %>% ggplot(aes(log2FoldChange, corrected_prob)) + geom_point(alpha=0.1, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.1) + 
              geom_line(stat='smooth', method='loess', color='gray', alpha=0.4, size=1.5, aes(y=prob)) +
              xlab('LFC') + ylab('Corrected Probability') +
              theme_minimal() + ggtitle('LFC vs model probability by gene')

Probability and Module-Diagnosis correlation

The Probabilitys increased for modules with negative correlation and decreased for modules with positive correlation

module_score = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>%
               left_join(biased_predictions %>% dplyr::select(ID, gene.score), by='ID') %>%
               left_join(assigned_module, by = 'ID') %>%
               dplyr::select(ID, prob, corrected_prob, Module, MTcor) %>% 
               left_join(data.frame(MTcor=unique(dataset$MTcor)) %>% arrange(by=MTcor) %>% 
                         mutate(order=1:length(unique(dataset$MTcor))), by='MTcor')

ggplotly(module_score %>% ggplot(aes(MTcor, corrected_prob)) + 
         geom_point(color=module_score$Module, aes(id=ID, alpha=corrected_prob^4)) +
         geom_hline(yintercept=mean(module_score$corrected_prob), color='gray', linetype='dotted') + 
         geom_line(stat='smooth', method = 'loess', color='gray', alpha=0.5, size=1.5, aes(x=MTcor, y=prob)) +
         geom_smooth(color='gray', method = 'loess', se = FALSE, alpha=0.3) + theme_minimal() + 
         xlab('Module-Diagnosis correlation') + ylab('Corrected Probability'))

Conclusion

This bias correction makes bigger changes in the distribution of the probabilities than the post-processing one

In general, the performance metrics decrease, but this isn’t necessarily bad, since we knew part of the good performance of the model was because of the confounding factor related to mean level of expression, so it was expected for the performance of the model to decrease once we removed this signal

Saving results

write.csv(test_set, file='./../Data/RM_weighting_bias_correction.csv', row.names = TRUE)

Session info

sessionInfo()

## R version 3.6.3 (2020-02-29)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB.UTF-8        LC_COLLATE=en_GB.UTF-8    
##  [5] LC_MONETARY=en_GB.UTF-8    LC_MESSAGES=en_GB.UTF-8   
##  [7] LC_PAPER=en_GB.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] grid      stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] expss_0.10.2       kableExtra_1.1.0   MLmetrics_1.1.1    car_3.0-7         
##  [5] carData_3.0-3      ROCR_1.0-7         gplots_3.0.3       DMwR_0.4.1        
##  [9] caret_6.0-86       lattice_0.20-41    mgcv_1.8-31        nlme_3.1-147      
## [13] reshape2_1.4.4     ggpubr_0.2.5       magrittr_1.5       RColorBrewer_1.1-2
## [17] gridExtra_2.3      viridis_0.5.1      viridisLite_0.3.0  plotly_4.9.2      
## [21] knitr_1.28         forcats_0.5.0      stringr_1.4.0      dplyr_1.0.0       
## [25] purrr_0.3.4        readr_1.3.1        tidyr_1.1.0        tibble_3.0.1      
## [29] ggplot2_3.3.2      tidyverse_1.3.0   
## 
## loaded via a namespace (and not attached):
##  [1] colorspace_1.4-1     ggsignif_0.6.0       ellipsis_0.3.1      
##  [4] class_7.3-17         rio_0.5.16           htmlTable_1.13.3    
##  [7] fs_1.4.0             rstudioapi_0.11      farver_2.0.3        
## [10] prodlim_2019.11.13   fansi_0.4.1          lubridate_1.7.4     
## [13] xml2_1.2.5           codetools_0.2-16     splines_3.6.3       
## [16] jsonlite_1.7.0       pROC_1.16.2          broom_0.5.5         
## [19] dbplyr_1.4.2         compiler_3.6.3       httr_1.4.1          
## [22] backports_1.1.8      assertthat_0.2.1     Matrix_1.2-18       
## [25] lazyeval_0.2.2       cli_2.0.2            htmltools_0.4.0     
## [28] tools_3.6.3          gtable_0.3.0         glue_1.4.1          
## [31] Rcpp_1.0.4.6         cellranger_1.1.0     vctrs_0.3.1         
## [34] gdata_2.18.0         crosstalk_1.1.0.1    iterators_1.0.12    
## [37] timeDate_3043.102    gower_0.2.1          xfun_0.12           
## [40] openxlsx_4.1.4       rvest_0.3.5          lifecycle_0.2.0     
## [43] gtools_3.8.2         MASS_7.3-51.6        zoo_1.8-8           
## [46] scales_1.1.1         ipred_0.9-9          hms_0.5.3           
## [49] yaml_2.2.1           quantmod_0.4.17      curl_4.3            
## [52] rpart_4.1-15         stringi_1.4.6        highr_0.8           
## [55] foreach_1.5.0        checkmate_2.0.0      TTR_0.23-6          
## [58] caTools_1.18.0       zip_2.0.4            shape_1.4.4         
## [61] lava_1.6.7           matrixStats_0.56.0   rlang_0.4.6         
## [64] pkgconfig_2.0.3      bitops_1.0-6         evaluate_0.14       
## [67] labeling_0.3         recipes_0.1.10       htmlwidgets_1.5.1   
## [70] tidyselect_1.1.0     plyr_1.8.6           R6_2.4.1            
## [73] generics_0.0.2       DBI_1.1.0            pillar_1.4.4        
## [76] haven_2.2.0          foreign_0.8-76       withr_2.2.0         
## [79] xts_0.12-0           survival_3.1-12      abind_1.4-5         
## [82] nnet_7.3-14          modelr_0.1.6         crayon_1.3.4        
## [85] KernSmooth_2.23-17   rmarkdown_2.1        readxl_1.3.1        
## [88] data.table_1.12.8    ModelMetrics_1.2.2.2 webshot_0.5.2       
## [91] reprex_0.3.0         digest_0.6.25        glmnet_3.0-2        
## [94] stats4_3.6.3         munsell_0.5.0

Bias Correction Weighting Technique